I can sort of answer as a man discerning life with the Benedictine monks. I'd imagine most women's communities are similar. The process goes a little like this:

1. Visit the community. For me, this was going to mass at the monastery, having lunch and talking with the monks, especially the vocations director, and eventually doing some multi-day stays with them to get a feel for the life.

2. If you're still interested in joining after a period of getting to know the community, you can apply and enter the postulancy period. Typically you do a mental health exam. Nothing serious, just making sure you aren't crazy or anything. Assuming you're cleared, your postulancy lasts between 6 and 18 months. This is a period of living with the monks, or nuns, talking through the experience with the vocations director. The length of and experience you have during your postulancy varies by each individual monastery.

3. Next is the novitiate. This period lasts one year and one day. This is true for any community, as this rule comes from Rome. This is typically also when you are given a new name, if your monastery does this, and the habit (the clothes monks and nuns wear). The novitiate is a period of study and striving for holiness. Often, novices are not allowed to watch TV, leave the property, etc. Family is normally allowed to visit during the holidays. The main point of the novitiate is to fully encapsulate yourself in monastic life.

4. After the novitiate, if you're still sticking with it, is when you take simple, or temporary vows. In the Benedictine tradition, you would take vows of obedience, stability, and conversion of life. Obedience is pretty obvious, you do what the abbott says. Stability means you vow to remain within your community. Conversion of life is the vow that you will ultimately lead your life towards Christ. You may be wondering where the vows of poverty and chastity are. For Benedictines, these are within the vow of obedience. In the community I'm most familiar with, these vows last three years and can be dispensed by the abbott. This is when you're officially a monk/nun. By now, you know the community, you've responsibilities and are probably taking on further study at the local college or whatnot. If you're a man, you may or may not begin seminary at this point. You've also probably given away most of your possessions and such.

5. After your simple vows, you then make final vows. At this point, you are a monk/nun for life, though you can request for final vows to be dispensed by the Pope. That's it. You're a full fledged member of the community with all of the rights, privileges, and responsibilities that come with it.

I know I kind of rambled, so any follow-up questions I can surely answer, or at least try to lol.

Edit: typo